Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity

TL;DR

This paper extends neural PDE solvers to handle coupled and non-linear equations, offering a comprehensive evaluation and new techniques for scientific simulations involving forward and inverse problems.

Contribution

It introduces an extension of the CNF neural PDE solver to non-linear and coupled equations, along with self-tuning methods and extensive benchmarking.

Findings

Extended neural PDE solver to non-linear and coupled equations

Developed self-tuning techniques for improved accuracy

Evaluated methods on diverse benchmark problems

Abstract

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to coupled and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.